Emily Riehl in her book Category theory in context said that ( page 90, first paragraph) :
we prove the analogous formula for limits or colimits of diagrams valued in any category by reducing the proofs of these general results to the case of limits in $\mathrm{Set}$.This strategy succeeds because limits and colimits in a general locally small category are defined representably in terms of limits in the category of sets
Here the "analogous formula" refers to the expressing colimits by coequalizer and coproduct.
You could get the book from http://www.math.jhu.edu/~eriehl/context/
I have read the proof on page 97 in this book. It seems that what the proof is trying to do is firstly to embed the coeuqalizer diagram to the category $\mathrm{Set^C}$ and get an euqalizer diagram, and then to show that functor $\mathrm{lim_{J^{op}}C}(F(j),-)$ is also an equalizer of the two arrows in the euqalizer diagram, and finally conclude that the two equalizers are isomorphic, that is $\mathrm{lim_{J^{op}}C}(F(j),-) \cong \mathrm{C}(C,-)$, which proves that C indeed is the colimit.
Actually I felt confused about the description "reducing the proof to the case of limits in $\mathrm{Set}$." I can't convince myselt that this proof is by reducing, since it seems more like "using the results in case of $\mathrm{Set}$ to caculate an equalizer of two arrows in $\mathrm{Set^C}$, and then compare two equalizers to get an isomorphism between them"...I feel that I don't get the spirit the author want to convey, could you elaborate more on this?
By the way, I also couldn't find out the motivation of this proof.Why to use Yoneda embedding to carry an coequalizer diagram to an equalizer diagram? Just because we want to use $\mathrm{lim_{J^{op}}C}(F(j),-) \cong \mathrm{C}(C,-)$ to complete the proof? Oh this is hard to understand for me...I'd appreciate if you could give me some motivation of this weird(to me) proof...